As a response to the somewhat discouraging circumstance that pupils in general are dissatisfied with mathematics teaching, research suggests some promising solutions for teachers’ inquiry based learning about inquiry based teaching. Yet, the auspicious attempts in the West to develop and implement inquiry based education have generally had little impact. As opposed to Western efforts, Japanese efforts to both develop and implement inquiry based teaching have proven impactful and lasting. This is often attributed to lesson study, a research-like format for teachers to inquire into pupils’ learning, with the aim of improving knowledge and skills to design and conduct lessons. Through lesson study, Japanese teachers and teacher educators devised open approach method, which provide guiding principles for inquiry based teaching. This thesis investigates the hypothesis, that lesson study and the guiding principles of open approach method can also help Danish teachers to develop knowledge and techniques of inquiry based teaching. For the design and analysis of experiments, theoretical tools derived from the Theory of Didactical Situations are employed. The tools comprise models to analyse didactical situations: pupils’ interplay with the didactical milieu (the mathematical problem and pertinent resources to solve it) and the teachers management of that interplay, and paradidactical situations: teachers’ interplay with the paradidactical milieu (the teaching problem and the didactical situation hypothesised to solve it).

In three articles, I investigate research questions concerning:

1) which conditions in the milieu are significant to realise principles of open approach method: From the analysis of one research lesson, four promoting conditions in the milieu and one hindering condition in the teachers’ traditional management of situation are identified. This also suggests, that even when good problems and materials for inquiries are developed, managing pupils’ inquiries is far from trivial.

2) what teachers learn in lesson studies and how that learning emerges and evolves: From the analysis of teachers’ paradidactical activities, specific teacher learnings are identified, and exemplary analyses of emergence and evolution are illustrated. The results suggest that learnings are strongly related to teachers’ anticipation of didactical situations, and that interactions leading to learning are interdependent.

3) how teachers’ anticipation of didactical situations evolve: From the analysis of pre-didactical situations (prior to the research lesson) of teachers’ first and third lesson study it is illustrated how teachers’ paradidactical practice evolved in step with a quantitative and qualitative increase in their anticipation of didactical situations.

In addition the thesis includes 4), a co-authored chapter proposing theoretical arguments for and tools to theorising lesson studies.